Orthogonality Properties of the Hermite and Related Polynomials G.Dattoli A, H.M

View metadata, citation and similar papers at core.ac.uk brought to you by CORE

provided by Elsevier - Publisher Connector

Journal of Computational and Applied Mathematics 182 (2005) 165–172 www.elsevier.com/locate/cam

Orthogonality properties of the Hermite and related polynomials G.Dattoli a, H.M. Srivastavab,∗, K.Zhukovsky c

aGruppo Fisica Teorica e Matematica Applicata (Unità Tecnico Scientiﬁca Tecnologie Fisiche Avanzante), ENEA-Centro Richerche Frascati Via Enrico Fermi 45, I-00044 Frascati, Rome, Italy bDepartment of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3P4, Canada cFaculty of Physics, Moscow State University, Vorobjovy Gory, Moscow 119899, Russian Federation Received 27 June 2004

Abstract The authors present a general method of operational nature with a view to investigating the orthogonality properties of several different families of the Hermite and related polynomials.In particular, the classical Hermite polynomials and some of their higher-order and multi-index generalizations are considered here. © 2004 Elsevier B.V. All rights reserved.

MSC: primary 33C45

Keywords: Orthogonality property; Hermite polynomials; Hermite–Kampé de Fériet (HKdF) polynomials; Multi-index Hermite polynomials; Gauss–Weierstrass transform; Generating functions; Laguerre polynomials; Operational identities; Exponential operators; Integral formulas; Biorthogonal functions; Higher-order Hermite (or Gould–Hopper) polynomials

1. Introduction, deﬁnitions and preliminaries

The so-called Hermite–Kampé de Fériet (HKdF) polynomials (or, alternatively, the two-variable Hermite polynomials) [1, p.341, Eq.(23)] : [n/ ] 2 xn−2r yr Hn(x, y) := n! (1) (n − 2r)!r! r=0

∗ Corresponding author.Tel.:+1 250 721 7455; fax: +1 250 721 8962. E-mail addresses: [email protected] (G.Dattoli), [email protected] (H.M. Srivastava), [email protected] (K.Zhukovsky).

can be defined by means of the following operational rule (cf. [2]): 2 j n Hn(x, y) = exp y {x }, (2) jx2 which was applied by Dattoli et al. [3] in order to reconsider the orthogonality property of the classical Hermite polynomials from a different point of view.It was indeed shown there [3] that such properties can be derived from fairly straightforward identities, in a direct way, by employing a few elementary properties of the exponential operators. In terms of the classical Hermite polynomials Hn(x) or Hen(x), it is easily seen from the definition (1) that H ( x,− ) = H (x) H (x, − 1 ) = (x). n 2 1 n and n 2 Hen (3) Furthermore, even though there exists the following close relationship [1, p.341, Eq.(21)] : x x n n/2 i n n/2 Hn(x, y) = (−i) y Hn √ = i (2y) Hen √ (4) 2 y i 2y with the classical Hermite polynomials, yet the usage of a second variable (parameter) y in the Hermite– Kampé de Fériet polynomials Hn(x, y) is found to be convenient from the viewpoint of their applications. Indeed, from an entirely different viewpoint and considerations, Hermite polynomials of several variables are introduced and investigated by Erdélyi et al. [5, p.283 et seq.]. Limiting ourselves to negative values of the variable y, which is treated in the present context as a parameter, we consider the following polynomial expansion: ∞ F(x)= an Hn(x, −|y|). (5) n=0 Then our goal will be that of specifying the coefficients an by the use of the operational representation (2) and of the formalism associated with the aforementioned operators.It is easily seen from Eq.(2) that 2 ∞ j n exp |y| {F (x)}= an x . (6) jx2 n=0 Our problem has, therefore, been reduced to that of finding the Taylor expansion of the function j2 (x) = exp |y| {F(x)}, (7) jx2 which, in view of the well-known Gauss–Weierstrass transform, can be written in the following form: 1 ∞ (x − )2 (x) = √ exp − F() d. (8) 2 |y| −∞ 4|y|

By recalling that the polynomials Hn(x, y) are speciﬁed by means of the generating function: ∞ tn H (x, y) = (xt + yt2), n! n exp (9) n=0 G. Dattoli et al. / Journal of Computational and Applied Mathematics 182 (2005) 165–172 167 we ﬁnd that ∞ 1 xn ∞ 1 2 (x) = √ Hn , − exp − F() d, (10) 2 |y| n! −∞ 2|y| 4|y| 4|y| n=0 which, when compared with Eq.(6), yields 1 ∞ 1 2 an = √ Hn , − exp − F() d. (11) 2 · n! |y| −∞ 2|y| 4|y| 4|y| This last identity (11) can be suitably exploited to conclude that the functions 1 x 1 x2 n(x) = √ Hn , − exp − (12) 2 · n! |y| 2|y| 4|y| 4|y| are biorthogonal to the polynomials Hn(x, −|y|). The corresponding expansion in series of the classical Hermite polynomials Hn(x) is easily obtained |y|= 1 from the above results by setting 2 . In these introductory remarks, we have shown how the use of operational tools has allowed the derivation of the orthogonality properties of the Hermite polynomials in a fairly direct way.In the following sections, we will show that the method developed here can be extended appropriately to more involved families of Hermite polynomials as well.

2. Multi-index Hermite polynomials and associated biorthogonal functions

Multi-variable and multi-index Hermite polynomials were introduced by Charles Hermite (1822–1901) himself in his memoirs dated 1864 in which he also investigated the relevant orthogonality properties (cf., e.g., [1, p.331 et seq.]).In this section, we will not follow the original treatment also exploited in [1], but the operational formalism of [2], which will provide a fairly direct understanding of the problem. According to the operational deﬁnition, the two-variable and two-index Hermite polynomials can be deﬁned as follows: 2 2 2 j j j m n Hm,n(x, ; y,|) = exp + + {x y }, (13) jx2 jxjy jy2 which provides the relevant series expansion:

(m,n) min s H (x, ; y,|) = m! n! H (x, )H (y, ) m,n s!(m − s)!(n − s)! m−s n−s (14) s=0 and the generating function

∞ um vn H (x, ; y,|) = (xu + u2 + yv + v2 + uv). m! n! m,n exp (15) m,n=0 168 G. Dattoli et al. / Journal of Computational and Applied Mathematics 182 (2005) 165–172

Let us now consider a two-variable function F(x,y) with the following expansion: ∞ F(x,y) = am,n Hm,n(x, −||; y,−|| −||). (16) m,n=0

We will evaluate the coefﬁcients am,n of expansion (16) by following the same procedure as before. Indeed, by using the operational representation (13), we get j2 j2 j2 (x, y) = exp || +|| +|| {F(x,y)} jx2 jxjy jy2 ∞ m n = am,n x y . (17) m,n=0 The problem now is that of specifying the action of the exponential operator: j2 j2 j2 E = exp || +|| +|| . (18) jx2 jxjy jy2 Thus we must look for a generalized version of the Gauss–Weierstrass transform used in the derivation of (8) above.By tacitly assuming, for convenience, that min(, , )>0, we can omit absolute values and rewrite the exponential operator E in (18) as follows: j 1 j 2 j2 2 E = exp + + = y − . (19) jx 2 jy jy2 4 By using the following integral formula: 1 ∞ √ exp( cˆ2) = √ exp −2 + 2 cˆ d, (20) −∞ we ﬁnd from (19) that 1 ∞ ∞ √ j 1 j j E := d d exp − 2 + 2 − 2 + − 2 . (21) −∞ −∞ jx 2 jy jy

Now the action of the exponential operator E on the function F(x,y) is easily derived by the use of the well-known identity j j exp a + b {F(x,y)}=F(x + a,y + b), (22) jx jy which, along with the deﬁnition (21), yields 1 ∞ ∞ √ EF (x, y) = d d exp − 2 + 2 F x + 2 ,y+ √ + 2 . (23) −∞ −∞ G. Dattoli et al. / Journal of Computational and Applied Mathematics 182 (2005) 165–172 169

Finally, upon performing the change of variables given by √ x + 2 = and y + √ + 2 = , (24) we obtain the two-variable extension of the Gauss–Weierstrass transform as follows: 1 ∞ ∞ EF (x, y) = √ d d 4 −∞ −∞ 1 · exp − (x − )2 − (x − )(y − ) + (y − )2 F(, ). (25) 4 Consequently, the use of the generating function (15) leads us to the following explicit expression for the coefﬁcients am,n in (16): ∞ ∞ − − a 1 1 2 2 am,n = √ d d Hm,n , − ; , − − 4 m! n! −∞ −∞ 2 4 2 4 4 1 · exp − 2 − + 2 F( , ), (26) 4 which, while generalizing the previous result, allows us to conclude that the functions x − y y − x 1 2 2 m,n(x, y) = √ Hm,n , − ; , − 4 · m! n! 2 4 2 4 4 1 · exp − x2 − xy + y2 (27) 4 are biorthogonal to the polynomials Hm,n(x, −||; y,−|| −||).

In the above case, too, the statement relevant to the orthogonality has been obtained, in a fairly direct way, by using the formalism associated with the exponential operator algebra.

3. Higher-order Hermite polynomials

In this section, we will discuss further generalizations of the orthogonality properties investigated in the preceding sections. (m) The higher-order Hermite polynomials Hn (x, y) (or, equivalently, the Gould–Hopper polynomials m gn (x, y) [9, p.76, Eq.1.9(6)] ) are given explicitly in [6, p.58, Eq.(6.2)]

[n/m] xn−mr yr H (m)(x, y) := n! =: gm(x, y), n (n − mr)! r! n (28) r=0 170 G. Dattoli et al. / Journal of Computational and Applied Mathematics 182 (2005) 165–172 where m is the order of the polynomial.The alternative operational deﬁnition: m (m) j n H (x, y) := exp y {x } (29) n jxm is particularly useful in (for example) deriving the following generating function: ∞ tn H (m)(x, y) = (xt + ytm). n! n exp (30) n=0 It is easily observed from the deﬁnition (28) that

(1) n (2) Hn (x, y) = (x + y) and Hn (x, y) = Hn(x, y), (31) where Hn(x, y) denotes the Hermite–Kampé de Fériet polynomials defined by (1).Indeed, just as it was pointed out by Srivastava and Manocha [9, pp.76–77] in connection with the Gould–Hopper polynomials m (m) gn (x, y), many obvious variations and special cases of the higher-order Hermite polynomials Hn (x, y) have been rediscovered in several different contexts, not only in the mathematical and statistical sciences, but also in the physical and engineering sciences (see also [2,7,8]). (m) The family of functions biorthogonal to this last family of polynomials Hn (x, y) have already been studied in [7,8], and subsequently (with a different technique) in [4].Here we will address the problem by following the technique developed in the preceding sections.Therefore, our starting point will be the assumption that the following expansion formula exists (see, for details [7,8]): ∞ (2q) q F(x)= an Hn x,(−1) |y| , (32) n=0 which, for q = 1, corresponds obviously to the polynomial expansion (5) above. We will now determine the expansion coefficients an in (32) from the identity: 2q q+ j (x|q) = exp (−1) 1|y| {F(x)} jx2q ∞ n = an x , (33) n=0 where we have made use of the operational definition (29) with m = 2q and y replaced by (−1)q|y|.The existence of the analogue of the Gauss–Weierstrass transform, which is needed in this case, is ensured by the following integral formula (cf. [4,7,8]): ∞ q+1 2q exp (−1) cˆ = exp(−ˆc )Sq( , ) d , (34) −∞ where, for convenience, ∞ 1 2q Sq( , ) = exp − u + iu du = Sq(− , ). 2 −∞ G. Dattoli et al. / Journal of Computational and Applied Mathematics 182 (2005) 165–172 171

Thus, in view of (34), Eq.(33) readily yields ∞ (x|q) = Sq( , |y|)F (x − ) d . (35) −∞ By performing the change of variables given by x − = ,

where the superscript n denotes the nth-order derivative of Sq( , |y|) with respect to . By comparing (32) with (36), it is clear that the functions

(−1)n+1 (x|q) = S(n)(x, |y|) n n! q (37) are biorthogonal to the polynomials (2q) q Hn x,(−1) |y| .

This result is in agreement with the analogous conclusions of [4,7,8], which were obtained within a different framework.

4. Concluding remarks and observations

In the preceding sections, we have exploited a general procedure to deal with the orthogonality properties of a large body of Hermite polynomial families.The obtained results and the fairly straightforward underlying formalism are elements proving the usefulness and the generality of the method which can easily be extended to other families of polynomials.The two-variable simple Laguerre polynomials can indeed be deﬁned by means of the following operational identity (cf. [2]): n n r n−r j j (−x) (−x) y n x Ln(x, y) := exp −y x = n! = y Ln , (38) jx jx n! (n − r)!(r!)2 y r=0 which can, in turn, be used to state the relevant orthogonality properties by applying a procedure analogous to that outlined here for the Hermite case.Here, as usual,

(0) −n Ln(x) := Ln (x) = y Ln(xy,y) = Ln(x, 1) denotes the classical Laguerre polynomial of order 0 and degree n in x. 172 G. Dattoli et al. / Journal of Computational and Applied Mathematics 182 (2005) 165–172

The extension of the above-detailed method to more involved forms of Hermite polynomials with n variables and n indices, such as those deﬁned below:   n n n n j2 j2 j2   nk H{n}({x}, {} {}) = exp j + l,j + j xk jx2 jxj jxl jx2 j=1 j lorthogonal polynomials by following the technique discussed in this paper.

Acknowledgements

The present investigation was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353.

References

[1] P.Appell, J.Kampé de Fériet, Fonctions Hypergéométriques et Hypersphériques; Polynômes d’Hermite, Gauthier-Villars, Paris, 1926. [2] G.Dattoli, Generalized polynomials, operational identities and their applications, J.Comput.Appl.Math.118 (2000) 111–123. [3] G.Dattoli, B.Germano, P.E.Ricci,Comments on monomiality, ordinary polynomials and associated bi-orthogonal functions, Appl.Math.Comput.154 (2004) 219–227. [4] G.Dattoli, B.Germano, P.E.Ricci,Higher order Hermite polynomials, associated bi-orthogonal functions and generalized heat equations, Integral Transform.Spec.Funct.16 (2005). [5] A.Erdélyi, W.Magnus, F.Oberhettinger, F.G.Tricomi, Higher Transcendental Functions, vol.II, McGraw-Hill Book Company, New York, Toronto and London, 1953. [6] H.W. Gould, A.T. Hopper, Operational formulas connected with two generalizations of Hermite polynomials, Duke Math. J.29 (1962) 51–63. [7] D.T.Haimo, C.Markett, A representation theory for solutions of a higher-order heat equation.I, J.Math.Anal.Appl.168 (1992) 89–107. [8] D.T.Haimo, C.Markett, A representation theory for solutions of a higher-order heat equation.II, J.Math.Anal.Appl.168 (1992) 289–305. [9] H.M. Srivastava, H.L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), Wiley, New York, Chichester, Brisbane and Toronto, 1984.